How to make a mosaic plane
Modifiers:

This is possible with the displace modifier, and you can use a procedural texture for the voronoi diagram. This way you don't even need to paint in each part of the voronoi texture, because they will generated each slightly darker/lighter.


Add a subsurf modifier to increase the mesh density.
Add the displace modifier and add a new displace texture (click New on the texture input of the displace modifier).
Set the Type of the texture to Voronoi and the Coloring to Position:


Add a Decimate modifier set to Planar to remove the unneeded geometry.


Result: (click for full size)



And here is an example blendfile.

You can sort of smooth out the edges with a Smooth modifier:



Cell fracture:

Expanding on Mike Pan's answer, here is a workflow using the Cell fracture addon. This addon is bundled with blender, but you still need to enable it in CtrlAltU> User Preferences > Addons:



Once it is enabled, 


Add a plane and subdivide it (Edit mode (Tab) > Select all (A) > W > Subdivide). The mesh density will affect the size of the resulting shards.


Select all and extrude down so it is now a manifold mesh:


With the cube selected in object mode, press Cell Fracture in 3D view > Tool shelf > Object tools > Cell fracture and set the Margin to 0. Changed settings are highlighted. Press OK when ready:


Delete the original cube, then select all the shards (making sure that one of them is active) and press CtrlJ to join them into one object:


Enter edit mode and remove double vertices by pressing A> Select all > W> Remove doubles. If needed increase the Merge Distance in the Tool Shelf > Redo menu:


Still with everything selected, run Delete> Limited Dissolve in order to further simplify the mesh (note that limited dissolve functions the same way as the Decimate modifier in planar mode):


Delete the excess geometry. The easest way to select it is by going into front orthographic view (Numpad 1, Numpad 5), the using Box select to only select the very top vertices:



You could also try going into top (Numpad 7) ortho view with solid shading enabled, then box selecting everything visible. This works, however you may need to deselect some vertices on the sides after doing this. 

The easiest way to do this is to enter front (Numpad 1) wire (Z) ortho view and deselect using box select. (B > MMB)
Invert the selection (CtrlI) and delete the selected vertices (Delete> Vertices). This should result in a nice clean plane of ngons, like this:


Select all (A) and press AltE> Individual Faces:


Without changing the selection, set the Pivot Point to Individual Origins (Ctrl. (period), or set it in 3D view > Header) and scale (S) to taper the extrusions slightly:


Invert the selection (CtrlI) and hide those vertices (H), so you are left with this:

 
Switch to face select mode (CtrlTab> Face) and select random (* 3D view > Header > Select*):


Move the randomly selected faces on the Z axis (GA) arbitrarily, randomly select again, move again, etc.

This process can be made slightly faster by running Select random from the Operator History (F3)
Unhide the hidden geometry (AltH). This should result in something like this: